postquam is a conjunction and praetereā an adverb, so it can't be "after which things" and "besides this one thing" as they would be respectively rendered with prepositions: post quae/haec and praeter hoc unum.

postquam goes with the verb: "After he had said these [words/things] ..."

Your first translation of the second sentence was correct. As you guessed, monēre can take a double accusative. In his Latin primer, Kennedy says (208, Note 3.):

moneō, advise, accūsō, arguō, accuse, cōgō, compel, take Acc. of the Thing if it is a Neuter Pronoun:Hoc tē moneō: I give you this advice.